For hp-finite element discretizations of time-harmonic Maxwell equations with Nédélec edge elements, the classical residual-based a posteriori error estimators have reliability and efficiency constants that grow algebraically with the local polynomial degree. Bürg (2012) proved reliability and efficiency for such an estimator and identified the standing open problems: obtain degree-uniform (hp-robust) bounds on general nonaffine element mappings, together with fully element-local efficiency for the face-jump terms, weakened solution-regularity assumptions, and a theory covering violation of the divergence condition. Subsequent work by Chaumont-Frelet, Ern, and Vohralík resolved the polynomial-degree-robustness question on affine simplicial meshes through equilibrated-flux and reconstruction estimators, culminating in an asymptotically constant-free estimator for the time-harmonic problem; the residual-based route remains non-robust in the degree. The open core posed here is the completion of polynomial-degree-robust a posteriori error estimation for Maxwell hp discretizations at the source's geometric generality: nonaffine, curved, and hexahedral element mappings — the geometries on which practical hp-Maxwell codes actually run — together with fully cellwise-local efficiency bounds, minimal-regularity reliability, and coverage of the constraint-violating regime.

Background

Time-harmonic Maxwell's equations describe electromagnetic fields oscillating at a fixed frequency; after eliminating the magnetic field, the electric field u\boldsymbol u satisfies a curl-curl problem of the form curl⁡(α curl⁡u)+βu=f\operatorname{curl}(\alpha\,\operatorname{curl}\boldsymbol u)+\beta\boldsymbol u=\boldsymbol f in a domain Ω\Omega, together with the Gauss-law constraint div⁡(βu)=0\operatorname{div}(\beta\boldsymbol u)=0 and suitable boundary conditions; the coefficients α,β\alpha,\beta encode the material's magnetic permeability and electric permittivity. The natural variational space is H(curl⁡;Ω)={v∈L2(Ω):curl⁡v∈L2(Ω)}\boldsymbol H(\operatorname{curl};\Omega)=\{\boldsymbol v\in\boldsymbol L^{2}(\Omega):\operatorname{curl}\boldsymbol v\in\boldsymbol L^{2}(\Omega)\}, discretized by Nédélec edge elements — finite element subspaces of H(curl⁡)\boldsymbol H(\operatorname{curl}) whose degrees of freedom are associated with mesh edges and which preserve the differential-complex (de Rham) structure that curl⁡\operatorname{curl} and div⁡\operatorname{div} require. In the hp-version of the finite element method (hp-FEM), both the element size hKh_{K} and the local polynomial degree pKp_{K} vary from element to element, so an estimator whose constants grow with pKp_{K} degrades precisely on the high-order elements the method is meant to exploit.

An a posteriori error estimator is a computable quantity η\eta, built from the data and the discrete solution alone, that bounds the unknown error: reliability, ∥u−uh∥≤Crel η\lVert\boldsymbol u-\boldsymbol u_{h}\rVert\le C_{\mathrm{rel}}\,\eta, certifies the error from above, while (local) efficiency, ηK≤Ceff∥u−uh∥patch(K)\eta_{K}\le C_{\mathrm{eff}}\lVert\boldsymbol u-\boldsymbol u_{h}\rVert_{\mathrm{patch}(K)}, prevents the estimator from drastically overestimating and makes it usable to mark elements for refinement. Residual-based estimators measure elementwise equation residuals and jumps of tangential field components across faces, and their analysis for Maxwell problems relies on testing the residual against element bubble functions (nonnegative element-local polynomials that vanish on the element boundary) and on H(curl⁡)\boldsymbol H(\operatorname{curl})-conforming interpolation operators. Bürg (2012) established such reliability (assuming extra regularity u∈Hr(curl⁡)\boldsymbol u\in\boldsymbol H^{r}(\operatorname{curl}), r>12+εr>\tfrac12+\varepsilon, to handle the interpolant) and efficiency for the hp-discretization, but with constants growing algebraically in pKp_{K}; the paper notes that inserting smoothing factors — as is possible for scalar elliptic problems, where Karkulik and Melenk (2015) later constructed hp-Clément-type quasi-interpolation (approximation operators built from local averaging over element patches, requiring less regularity than pointwise interpolation) by local high-order regularization — would remove this growth, but that evaluating such smoothing under general nonaffine element mappings FK:K^→KF_{K}:\hat K\to K is nontrivial and was not carried out.

An alternative estimator family sidesteps the residual route: equilibrated (reconstruction) estimators build discrete fields satisfying the divergence and curl constraints exactly, by solving small local minimization problems on patches, and then apply a Prager–Synge-type identity (a two-sided energy identity comparing the exact solution, the discrete solution, and any constrained reconstruction) to obtain reliable and efficient estimates with explicitly computable constants. Along this route, Chaumont-Frelet, Ern, and Vohralík proved polynomial-degree-robust H(curl⁡)\boldsymbol H(\operatorname{curl})-stability of discrete minimization on tetrahedra and built estimators for the curl-curl problem that are reliable, locally efficient, and degree-robust; Chaumont-Frelet and Vohralík constructed a stable local commuting projector with optimal hp approximation in H(curl⁡)\boldsymbol H(\operatorname{curl}); and Chaumont-Frelet (2026) reached an estimator for the time-harmonic problem whose reliability constant tends to one under mesh or degree refinement and whose efficiency constant is independent of pp, with low-frequency robustness — the strongest result to date. These p-robust (degree-robust) equilibrated estimators have also driven hp-adaptive algorithms in the elliptic case (Dolejší, Ern, Vohralík 2016).

All of these degree-robust results, however, live on shape-regular (elements of uniformly bounded aspect ratio) affine simplicial (tetrahedral) meshes. On nonaffine hexahedral elements the mapped Nédélec spaces are no longer polynomial, so the polynomial extension and discrete-minimization tools underlying the simplicial theory do not transfer; on curved isoparametric elements the pullbacks carry Jacobian-dependent weights that break the exactness properties the proofs use. Meanwhile, practical hp-Maxwell computation runs on exactly these geometries — hp-hexahedral Nédélec codes with hanging nodes (mesh vertices where elements of different refinement levels meet) are standard in electromagnetic simulation (e.g., the software work of Kinnewig, Wick, and Beuchler). The residual-based estimators still used for such problems retain degree-dependent efficiency constants, as restated in the current literature. This is the gap the present problem addresses.

Problem Statement

Construct a computable a posteriori error estimator for hp-Nédélec finite element discretizations of Maxwell-type problems (curl-curl problems with the Gauss-law constraint div⁡(βu)=0\operatorname{div}(\beta\boldsymbol u)=0) on general hp meshes whose elements may be nonaffine images of the reference element — including curved isoparametric and hexahedral elements — and prove, in this geometric generality:

  1. Reliability: ∥u−uh∥H(curl⁡)≤Crel η\lVert\boldsymbol u-\boldsymbol u_{h}\rVert_{\boldsymbol H(\operatorname{curl})}\le C_{\mathrm{rel}}\,\eta with CrelC_{\mathrm{rel}} independent of the local polynomial degrees pKp_{K} (or at worst polylogarithmic in them), under solution-regularity assumptions weaker than the u∈Hr(curl⁡)\boldsymbol u\in\boldsymbol H^{r}(\operatorname{curl}), r>12+εr>\tfrac12+\varepsilon, required by Bürg's (2012) residual analysis.
  2. Local efficiency: ηK≤Ceff∥u−uh∥patch(K)\eta_{K}\le C_{\mathrm{eff}}\lVert\boldsymbol u-\boldsymbol u_{h}\rVert_{\mathrm{patch}(K)} with CeffC_{\mathrm{eff}} independent of the polynomial degrees, and fully element-local (cellwise) efficiency bounds for the face-jump (tangential-trace jump) contributions; a resolution must establish both, the patch-local bound being the level achieved on affine simplicial meshes and the cellwise bound the sharper form the source states as open.
  3. Computability: the estimator must be computable from the data and the discrete solution alone, with the assumptions under which the stated bounds hold clearly identified.

The degree-uniform bounds must be established on the nonaffine geometry class rather than on affine simplices.

Two constituent questions from the source must be addressed within the same framework. First, the regime in which the divergence condition fails (div⁡(βu)≠0\operatorname{div}(\beta\boldsymbol u)\neq0): the source proposed only a numerically motivated modification of the estimator without proofs, and a complete resolution provides the estimator modification and the proofs covering this regime. Second, the locality level and the regularity assumptions actually achieved must be established rather than assumed: in particular, cellwise-local efficiency for the face-jump terms must be proven.

A resolution consists of the estimator construction with its transformation rules under nonaffine mappings, complete proofs of the stated bounds in this geometric generality, the estimator modification and proofs covering the regime in which the divergence condition fails (div⁡(βu)≠0\operatorname{div}(\beta\boldsymbol u)\neq0), and a demonstration (numerical or constructive) of computability. It is not solved by results restricted to affine simplicial meshes (the current state of the art) or by degree-dependent bounds.

References

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